As an illustration, we now apply the derived expressions to a step-index fiber, whose solutions are well known and well studied [1]}:
\(O_{l,m}(r,\theta )=\frac{1}{\sqrt{N_{l,m}}}J_l(p_{l,m}r)e^{il\theta } \quad for\ r\le a \nonumber \\=\frac{1}{\sqrt{N_{l,m}}}\frac{J_l(p_{l,m}a)}{K_l(q_{l,m}a)}K_l(q_{l,m}r)e^{il\theta } \quad for \ r\ge a,\) 
